# std::legendre, std::legendref, std::legendrel

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
float legendre ( unsigned int n, float x ); double legendre ( unsigned int n, double x ); long double legendre ( unsigned int n, long double x ); |  | (since C++17) (until C++23)
/* floating-point-type */ legendre( unsigned int n, /* floating-point-type */ x ); |  | (since C++23)
float legendref( unsigned int n, float x ); | (2) | (since C++17)
long double legendrel( unsigned int n, long double x ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double legendre ( unsigned int n, Integer x ); | (A) | (since C++17)

1-3) Computes the unassociated Legendre polynomials of the degree n and argument x. The library provides overloads of `std::legendre` for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

n | - | the degree of the polynomial
x | - | the argument, a floating-point or integer value

### Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

1
2n n!

dn
dxn

(x2
-1)n
, is returned.

### Error handling

Errors may be reported as specified in math_errhandling.

- If the argument is NaN, NaN is returned and domain error is not reported
- The function is not required to be defined for |x|>1
- If n is greater or equal than 128, the behavior is implementation-defined

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

The first few Legendre polynomials are:

Function | Polynomial
--- | ---
legendre(0, x) | 1
legendre(1, x) | x
legendre(2, x) |
1
2
(3x2 - 1)
legendre(3, x) |
1
2
(5x3 - 3x)
legendre(4, x) |
1
8
(35x4 - 30x2 + 3)

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::legendre(int_num, num) has the same effect as std::legendre(int_num, static_cast<double>(num)).

### Example

Run this code

    #include <cmath>
    #include <iostream>

    double P3(double x)
    {
        return 0.5 * (5 * std::pow(x, 3) - 3 * x);
    }

    double P4(double x)
    {
        return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
    }

    int main()
    {
        // spot-checks
        std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
                  << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
    }

Output:

    -0.335938=-0.335938
    0.157715=0.157715

### See also

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17) | Laguerre polynomials (function)
hermitehermitefhermitel (C++17)(C++17)(C++17) | Hermite polynomials (function)

### External links

Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource.

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